What is the value of (12)(pi/8)^2-2csc^2(pi/4) ?

The value of (12)(pi/8)^2-2csc^2(pi/4) is (3pi^2)/(16)-4

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(12)(pi/8)^2-2csc^2(pi/4)

Solution

(12) (\frac{π}{8})^{2} - 2 csc^{2} (\frac{π}{4})

Solution

\frac{3 π ^{2}}{16} - 4

Decimal Notation

- 2.14944 \dots

Solution steps

(12) (\frac{π}{8})^{2} - 2 csc^{2} (\frac{π}{4})

= 12 (\frac{π}{8})^{2} - 2 csc^{2} (\frac{π}{4})

Use the following trivial identity:

csc (\frac{π}{4}) = 2

csc (\frac{π}{4})

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= 2

= 12 (\frac{π}{8})^{2} - 2 (2)^{2}

Simplify

12 (\frac{π}{8})^{2} - 2 (2)^{2} : \frac{3 π ^{2}}{16} - 4

12 (\frac{π}{8})^{2} - 2 (2)^{2}

12 (\frac{π}{8})^{2} = \frac{3 π ^{2}}{16}

12 (\frac{π}{8})^{2}

(\frac{π}{8})^{2} = \frac{π ^{2}}{8 ^{2}}

(\frac{π}{8})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{π ^{2}}{8 ^{2}}

= 12 \cdot \frac{π ^{2}}{8 ^{2}}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π ^{2} \cdot 12}{8 ^{2}}

Factor

12 : 2^{2} \cdot 3

Factor

12 = 2^{2} \cdot 3

Factor

8^{2} : 2^{6}

Factor

8 = 2^{3}

= (2^{3})^{2}

Simplify

(2^{3})^{2} : 2^{6}

(2^{3})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= 2^{6}

= 2^{6}

= \frac{2 ^{2} \cdot 3 π ^{2}}{2 ^{6}}

Cancel

\frac{π ^{2} \cdot 2 ^{2} \cdot 3}{2 ^{6}} : \frac{3 π ^{2}}{2 ^{4}}

\frac{π ^{2} \cdot 2 ^{2} \cdot 3}{2 ^{6}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{2}}{2 ^{6}} = \frac{1}{2 ^{6 - 2}}

= \frac{3 π ^{2}}{2 ^{6 - 2}}

Subtract the numbers:

6 - 2 = 4

= \frac{3 π ^{2}}{2 ^{4}}

= \frac{3 π ^{2}}{2 ^{4}}

2^{4} = 16

= \frac{3 π ^{2}}{16}

2 (2)^{2} = 4

2 (2)^{2}

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

= \frac{3 π ^{2}}{16} - 4

= \frac{3 π ^{2}}{16} - 4

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Frequently Asked Questions (FAQ)

What is the value of (12)(pi/8)^2-2csc^2(pi/4) ?
The value of (12)(pi/8)^2-2csc^2(pi/4) is (3pi^2)/(16)-4